We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.
Keywords: shape memory alloys, thermo-mechanics, well-posedness, hysteresis operator
@article{M2AN_2010__44_6_1239_0,
author = {Krej\v{c}{\'\i}, Pavel and Stefanelli, Ulisse},
title = {Well-posedness of a thermo-mechanical model for shape memory alloys under tension},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1239--1253},
year = {2010},
publisher = {EDP Sciences},
volume = {44},
number = {6},
doi = {10.1051/m2an/2010024},
mrnumber = {2769056},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2010024/}
}
TY - JOUR AU - Krejčí, Pavel AU - Stefanelli, Ulisse TI - Well-posedness of a thermo-mechanical model for shape memory alloys under tension JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2010 SP - 1239 EP - 1253 VL - 44 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2010024/ DO - 10.1051/m2an/2010024 LA - en ID - M2AN_2010__44_6_1239_0 ER -
%0 Journal Article %A Krejčí, Pavel %A Stefanelli, Ulisse %T Well-posedness of a thermo-mechanical model for shape memory alloys under tension %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2010 %P 1239-1253 %V 44 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2010024/ %R 10.1051/m2an/2010024 %G en %F M2AN_2010__44_6_1239_0
Krejčí, Pavel; Stefanelli, Ulisse. Well-posedness of a thermo-mechanical model for shape memory alloys under tension. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 44 (2010) no. 6, pp. 1239-1253. doi: 10.1051/m2an/2010024
[1] , A model of 3D shape memory alloy materials. J. Math. Soc. Jpn. 57 (2005) 903-933. | Zbl
[2] , and , Modelling and numerical simulation of martensitic transformation in shape memory alloys. Contin. Mech. Thermodyn. 15 (2003) 463-485. | Zbl
[3] and , Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations. Int. J. Numer. Methods Eng. 55 (2002) 1255-1284. | Zbl
[4] and , A three-dimensional model describing stress-temperature induced solid phase transformations. Part I: Solution algorithm and boundary value problems. Int. J. Numer. Methods Eng. 61 (2004) 807-836. | Zbl
[5] and , A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: Thermomechanical coupling and hybrid composite applications. Int. J. Numer. Methods Eng. 61 (2004) 716-737. | Zbl
[6] and , A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite. Int. J. Non-Linear Mech. 32 (1997) 1101-1114. | Zbl
[7] , and , A phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity, in Topics on Mathematics for Smart Systems (Rome, 2006), World Sci. Publishing (2007) 1-14. | Zbl
[8] , and , A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int. J. Plast. 23 (2007) 207-226. | Zbl
[9] , and , A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Meth. Appl. Sci. 18 (2008) 125-164. | Zbl
[10] , and , A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties. Comput. Methods Appl. Mech. Eng. 198 (2009) 1631-1637.
[11] and , A three-dimensional model for magnetic shape memory alloys. Preprint IMATI-CNR 27PV09/20/0 (2009).
[12] and , Hysteresis and phase transitions, Applied Mathematical Sciences 121. Springer-Verlag, New York (1996). | Zbl
[13] , Global existence for the three-dimensional Frémond model of shape memory alloys. Nonlinear Anal. 24 (1995) 1565-1579. | Zbl
[14] and , Global existence for a three-dimensional model for the thermodynamical evolution of shape memory alloys. Nonlinear Anal. 18 (1992) 873-888. | Zbl
[15] , , Eds., SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference. ASM International (2003).
[16] , , and , Eds., Engineering aspects of shape memory alloys. Butterworth-Heinemann (1990).
[17] , Martensitic domain boundaries in shape-memory alloys as solitary waves. J. Phys. C4 Suppl. 12 (1982) 3-15.
[18] and , Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys. J. Phys. Condens. Matter 2 (1990) 61-77.
[19] , Matériaux à mémoire de forme. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 304 (1987) 239-244.
[20] , Non-smooth Thermomechanics. Springer-Verlag, Berlin (2002). | Zbl
[21] and , A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Methods Appl. Mech. Eng. 191 (2001) 215-238. | Zbl
[22] and , Shape memory behaviour: modelling within continuum thermomechanics. Int. J. Solids Struct. 40 (2003) 827-849. | Zbl
[23] , On uniqueness for evolution problems with hysteresis, in Mathematical Models for Phase Change Problems, J.F. Rodrigues Ed., Birkhäuser, Basel (1989) 377-388. | Zbl
[24] , and , Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys. Nonlinear Anal. 15 (1990) 977-990. | Zbl
[25] , Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Int. Series Math. Sci. Appl. 8. Gakkotosho, Tokyo (1996). | Zbl
[26] and , Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires. Preprint, IMATI-CNR, 12PV09/10/0 (2009).
[27] , , , , and , Shape memory alloys, Part II: Modeling of polycrystals. Mech. Materials 38 (2006) 391-429.
[28] , Thermomechanical theory of martensitic phase transformations in inelastic materials. Int. J. Solids Struct. 35 (1998) 889-940. | Zbl
[29] , The thermomechanics of plasticity and fracture, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992). | Zbl
[30] and , Thermally driven phase transformation in shape-memory alloys. Adv. Math. Sci. Appl. 17 (2007) 160-182. | Zbl
[31] , and , On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys. SIAM J. Math. Anal. 41 (2009) 1388-1414. | Zbl
[32] , , and , Error estimates for discretizations of a rate-independent variational inequality. WIAS Preprint n. 1407 (2009).
[33] , , and , Error control for space-time discretizations of a 3D model for shape-memory materials, in Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer (2009).
[34] , Three-dimensional model of thermomechanical evolution of shape memory materials. Control Cybernet. 29 (2000) 341-365. | Zbl
[35] , and , Macroscopic constitutive law for SMA: Application to structure analysis by FEM. Materials Sci. Eng. A 438-440 (2006) 454-458.
[36] and , A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite. Int. J. Plast. 23 (2007) 1679-1720. | Zbl
[37] and , RL models of pseudoelasticity and their specification for some shape-memory solids. Eur. J. Mech. A Solids 13 (1994) 21-50. | Zbl
[38] and , Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation. Int. J. Plast. 28 (2008) 455-482. | Zbl
[39] , Models of microstructure evolution in shape memory alloys, in Nonlinear Homogenization and its Appl. to Composites, Polycrystals and Smart Materials, P. Ponte Castaneda, J.J. Telega, B. Gambin Eds., NATO Sci. Series II/170, Kluwer, Dordrecht (2004) 269-304.
[40] , and , Three-dimensional model for solids undergoing stress-induced phase transformations. Eur. J. Mech. A Solids 17 (1998) 789-806. | Zbl
[41] , Analysis of a variable time-step discretization for the Penrose-Fife phase relaxation problem. Nonlinear Anal. 45 (2001) 213-240. | Zbl
[42] and , Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Phys. Solids 49 (2001) 709-737. | Zbl
[43] , , and , Implementation of a model taking into account the asymmetry between tension and compression, the temperature effects in a finite element code for shape memory alloys structures calculations. Comput. Materials Sci. 41 (2007) 208-221.
[44] , Differential Models of Hysteresis, Applied Mathematical Sciences 111. Springer, Berlin (1994). | Zbl
[45] , and , Quasi-linear thermoelasticity system arising in shape memory materials. SIAM J. Math. Anal. 38 (2007) 1733-1759. | Zbl
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